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In mathematics, the Hahn–Banach Theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting"

I need a short and excellent proof for Hahn–Banach theorem. Anyone could help me ?

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  • $\begingroup$ What's wrong with the proof in whichever book on functional analysis you're reading? $\endgroup$ – Daniel Fischer Dec 2 '14 at 20:35
  • $\begingroup$ the long detailed of proof. $\endgroup$ – Michle Jordan Dec 2 '14 at 20:37
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    $\begingroup$ Erm, sorry. Do I understand that right, the proof in your book is too long and detailed for you? How long is it? Since the theorem isn't quite trivial, the proof must take more than a couple of lines, but more than a couple of pages would be excessive. $\endgroup$ – Daniel Fischer Dec 2 '14 at 20:53
  • $\begingroup$ Dear @DanielFischer, would you please send me a correct link for see it's proof? $\endgroup$ – Michle Jordan Dec 2 '14 at 20:57
  • $\begingroup$ Dear Michle, can you tell me the main steps of your proof? The one that I know doesn't seems so long! $\endgroup$ – GGG Dec 2 '14 at 23:14
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This thesis contains a lot of different proofs of Hahn-Banach theorem and much more.

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